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Mirrors > Home > MPE Home > Th. List > ulmpm | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
ulmpm | ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmf 23940 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆)) | |
2 | uzssz 11583 | . . . 4 ⊢ (ℤ≥‘𝑛) ⊆ ℤ | |
3 | ovex 6577 | . . . . 5 ⊢ (ℂ ↑𝑚 𝑆) ∈ V | |
4 | zex 11263 | . . . . 5 ⊢ ℤ ∈ V | |
5 | elpm2r 7761 | . . . . 5 ⊢ ((((ℂ ↑𝑚 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ)) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) | |
6 | 3, 4, 5 | mpanl12 714 | . . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
7 | 2, 6 | mpan2 703 | . . 3 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
8 | 7 | rexlimivw 3011 | . 2 ⊢ (∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
9 | 1, 8 | syl 17 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ↑pm cpm 7745 ℂcc 9813 ℤcz 11254 ℤ≥cuz 11563 ⇝𝑢culm 23934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-pm 7747 df-neg 10148 df-z 11255 df-uz 11564 df-ulm 23935 |
This theorem is referenced by: ulmf2 23942 |
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